Logarithmic-kernel behavior of exceptional elliptic candidates

Determine whether exceptional elliptic candidates whose remaining algebraic one-form can be written as a logarithmic differential evaluate to iterated integrals with logarithmic kernels.

Background

The paper constructs rational basis candidates for elliptic Feynman integrals by matching algebraic one-forms on an elliptic curve to selected representatives of the first, second, and third kinds. In special cases, the elliptic obstruction disappears: either the algebraic contribution vanishes or it can itself be expressed as a logarithmic differential.

The authors explicitly leave unresolved whether such exceptional candidates lead, after integration, to iterated integrals whose kernels remain logarithmic. Resolving this would clarify when elliptic-looking integrands can still be incorporated into a polylogarithmic-style iterated-integral framework.

References

It would be interesting to understand whether they evaluate to iterated integrals with logarithmic kernels, which we leave for future work.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Section 3, immediately following Eq. (3.4), Section “Rational bases from elliptic leading singularities”